1 Introduction

We will here illustrate the use of a probabilistic approach to solving different types of problems. We start with one of the earliest applications.

1.1 The Buffon needle problem (1777)

Buffon considered a whole set of similar problems formulated as games. As a special case , Buffon in his own words states, "I assume that in a room, the floor of which is merely divided by parallel lines, a stick is thrown uppwards and one of the players bets the stick will not intersect any of the parallels on the floor, whereas on the contrary the other one bets the stick will intersect some one of these lines; it is required to find the chances of the two players. It is possible to play this game with a sewing needle or a headless pin."

He then demonstrates that for a fair game between two players, the ratio of the length of the needle, l, to the distance between the parallel lines, d, (d>l) must equal p/4 (p = 3.14159265... i.e., pi) for this provides the probability of an intersection equal to 1/2.

The probability of an intersection in the general case can easily be derived, see the pages 1.1-1.2 in the folder. Choosing l=d gives the probability P that the needle intersects a line

                          P = 2/p   

We may now write this as

                          p = 2/P

and estimate p by performing the needle experiment. Throw a needle with length equal to the distance between the parallel lines N times and record the number of times, n, that a line is intersected by the needle. Then n/N is an estimate of P and we may thus calculate p as

                          p ˜ 2N/n
 

The accuracy of the estimate 2N/n improves with increasing N.

This is an example of solving a deterministic problem by transforming it into a stochastic one.